Question

Question given below has a problem and two statements I & II. Decide if the information given in the statement is sufficient for answering the problem.

The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?
(1) The height of the equilateral triangle is equal to $6\sqrt{3}$
(2) The area of the equilateral triangle is equal to $36\sqrt{3}$.

• A. Data in statement I alone is sufficient.
• B. Data in statement II alone is sufficient.
• C. Data in both statements together is sufficient.
• D. Data in both statements together is not sufficient.
• E. Either of the statements I or II is sufficient.

Answer 1

The correct option is E Either of the statements I or II is sufficient.

No calculation is needed to solve this problem. Both equilateral triangles and squares are regular figures-those that can change size, but never shape.

Regular figures (squares, equilaterals, circles, spheres, cubes, 45-45-90 triangles, 30-60-90 triangles, and others) are those for which you only need one measurement to know every measurement. For instance, if you have the radius of a circle, you can get the diameter, circumference, and area. If you have a 45-45-90 or 30-60-90 triangle, you only need one side to get all three. In this problem, if you have the side of an equilateral, you could get the height, area, and perimeter. If you have the side of a square, you could get the diagonal, area, and perimeter.

If you have two regular figures, as you do in this problem, and you know how they are related numerically ("the side of an equilateral triangle has the same length as the diagonal of a square"), then you can safely conclude that any measurement for either figure will give you any measurement for either figure.

The question can be rephrased as, "What is the length of any part of either figure?"(1)This gives you the height of the triangle. SUFFICIENT.

(2)This gives you the area of the triangle. SUFFICIENT.

If you really wanted to "prove" that the answer is (D), you could waste a lot of time:

From statement 1, if the height of the equilateral is $6\sqrt{3}$, then the side equals 12, because heights and sides of equilaterals always exist in that ratio (the height is always one-half the side times $\sqrt{3}$). Then you would know that the diagonal of the square was also equal to 12, and from there you could use 12 the 45-45-90 formula to conclude that the side of the square was $\frac{12}{\sqrt{2}}$, and therefore that the area was $\frac{144}{2}$ or 72.

Similarly, from statement 2, you could conclude that if the area of the triangle is $36\sqrt{3}$, then the base times the height is $72\sqrt{3}$, and that since the side and height of an equilateral always exist in a fixed ratio (as above, the height is always one-half the side times $\sqrt{3}$), that the side is 12 and the height is $6\sqrt{3}$. Then, as above, you would know that the diagonal of the square was also equal to 12, and from there you could use the 45-45-90 formula to conclude that the side of the square was $\frac{12}{\sqrt{2}}$ and therefore that the area was $\frac{144}{2}$, or 72.

Who's got the time? This is a logic problem more than it is a math problem. If you understand the logic behind regular figures, you can answer this question in under 30 seconds with no math what so ever.

The correct answer is (D).